What’s the point of Physick for Summoner?

[Exodus-E]

Summoning was developed as a means to usurp primal power and then use it against other primals. Some primals have healing capacity, so nothing in Summoner excludes that. A summoner using a Lakshmi-egi or Moggle Mog-egi would have healing powers without breaking a single iota of summoner lore, as would a Summoner seeped with Phoenix aether (so all of them in modern Eorzea). There is no such lore that contradicts a summoner being able to use healing powers, and that's before considering that arcanima has that available.

FTFY (the official term is "arcanima").
Else, this is completely spot on.

As for Red Mages, they do not use modern conjury or thaumaturgy. They use a unique school of magic that follows very strict rules (the biggest being the only aether permitted to be used is personal aether, and waste aether from red magic) and both rules are broken by conjury and thaumaturgy.* The school was forged in isolation a long time ago, by mages attempting to deliberately learn how to Sylphie their white and black spells without suffering Sylphie-health-problems, and this was at the time VERY forbidden study because of that. The resulting magic carried a red tint to it (because it's cast from one's own health in the lore) and that's why it's called Red Magic. However, the practice of using white and black magic being forbidden is irrelevant because Red Magic was developed in secret and not within Gridania or Belah'diah, which is who forbid those practices. (There is no evidence to support or oppose Sharlayan forbidding those practices, for the record.)

1. To be precise, it was developed by the survivors of the 6th Umbral Calamity (mostly Amdapori and Mhachi), who set aside their differences in order to create a more "responsible" discipline in terms of aether consumption.
2. X'rhun did advise Arya to practice her conjury and thaumaturgy skills in order to get better at red magick. Ergo, they are compatible with it (if only to serve as a "starting point" of sorts).
3. White magick is currently restricted to the Padjal in Gridania while black magick used to be outright forbidden in Ul'dah until very recently (and even then, it's for "research purposes" and supervised by the Thaumaturge guild).